Absolute Value
Each real number can be regarded as unsigned number with a sign + or - as prefix or multiplier. The unsigned number provides the absolute value or magnitude of the real number. Along the real coordinate or number line, positive numbers and unsigned numbers are considered to be identical.
The magnitude or absolute value function (a computation rule)
The absolute value or magnitude of a positive number 5 = +5 is the number itself.
The absolute value or magnitude of zero 0 is zero 0.
The absolute value or magnitude of a negative number _7 is +7. The same result can be obtained by multiplying by -1 or computing the negative of _ 7 as 7 = (-1)( _ 7) = - ( _ 7)
The absolute value of a real number gives its distance to 0.
Exercise: Find the distance to zero (absolute value) of the following numbers: 5.6, 23.85, 3.14, p, -p, 10, -10, -45.67, 45.67
The computation of the absolute value of a number x can be described in words, the absolute value of x, that is,
|x| is given by the number x when x > 0 and by -x = (-1)x when x < 0
or equivalent by the shorthand formula
|
|x| = { |
x when x > 0, and |
| -x when x < 0. |
Some text may write if in place of when.
The first difficulty in dealing with the absolute value function f(x) = |x| lies in accepting the shorthand formula as the starting point for future computations and reasoning with absolute value.
Example I: Let us compute | -5|. Now x = -5 < 0. So according to the formula |x| = -x = -(-5) and the latter gives 5, the same result as dropping the sign in front of the number).
Example II: Let us compute |+2.5|. Here x = 2.5 > 0. So according to the formula |x| = x = 2.5 and we are done.
Another Viewpoint
Comparison of Distance to Origin
Distance to the origin of a coordinate system can be compared for points along a line, in a plane or in space.
Which number -5 or 3 has the greater distance to the origin? The answer is -5 as it is 5 units away from the origin while 3 is 3 units away.
The distance of a number to the origin is called is magnitude or absolute value. It can be obtained by dropping the sign to obtain an unsigned number.. It can be obtained by changing the sign to positive.
Here the number -5 is less than 3 in the linear order of the real number or coordinate line, but its magnitude (aka. distance to the origin or absolute value) is bigger or greater than that of 3.
a.k.a stands for also known as
The distance of a number x to the origin (a.k.a magnitude or absolute value) is given by the number x when x is positive or zero, and the negative of x, that is -x, when x is negative.
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